Regularity of a boundary having a Schwarz function.(English)Zbl 0728.30007

Summary: In his book, P. J. Davis discussed various interesting aspects concerning a Schwarz function. It is a holomorpic function S defined in a neighbourhood of a real analytic arc satisfying $$S(\zeta)={\bar \zeta}$$ on the arc, where $${\bar \zeta}$$ denotes the complex conjugate of $$\zeta$$. The author defines a Schwarz function for a portion of a boundary of an arbitrary open set and shows the regularity of the portion of the boundary. More precisely, let $$\Omega$$ be an open set of the unit disk B such that the boundary $$\partial \Omega$$ contains the origin 0 and let $$\Gamma =(\partial \Omega)\cap B$$. A function S defined on $$\Omega\cup \Gamma$$ is called the Schwarz function of $$\Omega\cup \Gamma$$ if (i) S is holomorphic in $$\Omega$$, (ii) S is continuous on $$\Omega\cup \Gamma$$ and (iii) $$S(\zeta)={\bar \zeta}$$ on $$\Gamma$$. The author gives a classification of a boundary having a Schwarz function. The main theorem asserts that there are four types of the boundary if 0 is not an isolated boundary point of $$\Omega$$ : 0 is a regular, nonisolated degenerate, double or cusp point of the boundary.

MSC:

 30C35 General theory of conformal mappings 30C99 Geometric function theory

Keywords:

free boundary; Schwarz function
Full Text:

References:

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